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>>16490
ax^2+bx+c=0
So you can see everything is on one side and on the other we got 0 so think about this like "where is the function equal to zero [f(x)=0]"
Look at the picture, you can see that there are 3 possibilities; 2 points at which function is equal to 0, 1 point (only the tip of parabola) or there are none. We solve it a bit differently in my country (I think in easier to understand form) so I'll show you how we are thought. First you look at 'a', a can't be equal to 0 (it would be linear equation if a=0), ok 'a' isn't 0 , that means we got a parabola, we can determine whether there are 2, 1 or 0 intersections with the X axis by calculating Δ (capital greek delta)
Δ=b^2 - 4ac
ok so you calculated the Δ, if the Δ<0 that means no intersections with X axis, Δ=0 1 intersection with X axis (the tip) and if Δ>0 we got 2 intersections with the X axis. When you got Δ<0 the equation can't be solved (no points at which function is equal to 0), if Δ=0 (only the tip is touching the X axis) you calculate the X coordinate of the tip, formula for that is (-b)/2a; if Δ>0 we must do a little bit more to solve the equation, because for Δ>0 there are 2 points touching the x axis, we use 2 formulas (-b+√Δ)/(2a) and (-b-√Δ)/(2a) similar to standard quadratic formula.
let's do some examples then
x^2 + 2x + 1 = 0
1) Δ= b^2 - 4ac = 4 - 4*1*1 = 0
2) Δ = 0 so we use formula for the X coordinate of the tip, (-b)/2a = (-2)/(2*1) = (-2)/2 = -1
3) f(x) = 0 if x = 1
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